On the crest factor for dissipative partial differential equations

Abstract

In this work, we have introduced and then computed the so-called crest factor associated with solutions of dissipative partial differential equations (PDEs). By taking two paradigmatic dissipative PDEs, we estimated in an explicit and accurate manner the values of the crest factor of their solutions. We then analysed and compared the estimates as a function of the positive parameter which appears in the PDEs in space dimensions one and two. These estimates shed some light on the dynamics of the fluctuations of the solutions of the two model PDEs, and therefore provide a criterion for discerning between small and large potential excursions in space for the solution of any dissipative PDE. Being able to detect between small and large intermittent fluctuations is one of the hallmarks of turbulence. We believe that the crest factor is an appropriate tool for extracting space fluctuation features in solutions of dissipative PDEs.